Phase Retrieval for Wide Band Signals

Phase-Retrieval in Shift-Invariant Spaces with Gaussian Generator

“…By construction, f + g and f − g are linearly independent, and thus sign-retrieval is not unique. w∈W − e 2πiz − e 2πiw 1 − e 2πiw (15) for some r ∈ Z and C ∈ C. The product converges uniformly on compact sets.…”

A note on the phase retrieval of holomorphic functions

“…Note that now f, g ∈ H 2 (D) so that we could apply [10,Lemma 4.5] and obtain g = cf , for some c ∈ T. We will give an alternative simpler proof.…”

“…Our aim in this paper is to generalize some uniqueness results on the phase retrieval of holomorphic functions on the unit disc. Indeed, in [10], we considered the phase retrieval problem: given f ∈ L 2 (R) such that its Fourier transform satisfies an exponential decay condition, find all functions g ∈ L 2 (R) such that |f | = |g| on R and with its Fourier transform satisfying the same exponential decay condition as f . Using a Paley-Wiener theorem and a conformal map, the problem can be translated to the Hardy space on the unit disc.…”

“…By the inner-outer factorization, the explicit form of the solution was obtained. One of our motivation stems from one of the coupled phase retrieval problems from [10,Lemma 4.5], which states that for f and g in the Hardy space on the unit disc with |f | = |g| on (−1, 1) and on some segment inside the disc, g can be obtained uniquely from f . For our first result, we extend this uniqueness result to holomorphic functions on open connected domains.…”

“….Perez III In the same spirit when ω is not open, we have shown in the paper[10, Lemma 4.5] that if f, g ∈ H 2 (D) and|f (z)| = |g(z)|, z ∈ (−1, 1) ∪ e iα (−1, 1)where α / ∈ πQ, then f and g are equal up to the multiplication of a unimodular constant. For this result, uniqueness was established by showing that the Blaschke products, singular inner parts, and outer parts of f and g are equal.…”